PHYSICAL REVIEW D, VOLUME 64, 083006 Primordial magnetic fields from metric perturbations Antonio L. Maroto CERN Theory Division, CH-1211 Geneva 23, Switzerland and Departmento de Fı´sica Teo´rica, Universidad Complutense de Madrid, 28040 Madrid, Spain ~Received 14 May 2001; published 24 September 2001! We study the amplification of electromagnetic vacuum fluctuations induced by the evolution of scalar metric perturbations at the end of inflation. Such perturbations break the conformal invariance of Maxwell equations in Friedmann-Robertson-Walker backgrounds and allow the growth of magnetic fields on super-Hubble scales. We relate the strength of the fields generated by this mechanism with the power spectrum of scalar perturba- tions and estimate the amplification on galactic scales for different values of the spectral index. Finally we discuss the possible effects of finite conductivity during reheating. DOI: 10.1103/PhysRevD.64.083006 PACS number~s!: 98.62.En, 98.80.Cq e o lle fe th in sa g , rs b f d he h ba is at c e ti e f on v- hi o he ca ng the ons e e ies rge ion y, tua- as ion d or- re- g. fore nce the ons he and by ced - Be- ia- ed etic ent a- and . In ing de- in lu- I. INTRODUCTION The existence of cosmic magnetic fields with large coh ence lengths~.10 kpc! and a typical strength of 1026 G still remains an open problem in astrophysics@1#. A partial explanation, widely considered in the literature, is based the amplification of seed fields by means of the so ca galactic dynamo mechanism. In this mechanism, the dif ential rotation of the galaxy is able to transfer energy into magnetic field, but nevertheless it still requires a preexisit field to be amplified. The present bounds on the neces seed fields to comply with observations are in the ran Bseed*10217210222 G (h50.6520.5) at decoupling time coherent on a comoving scale oflG;10 kpc, for a flat uni- verse without a cosmological constant. For a flat unive with a nonvanishing cosmological constant, the limits can relaxed up toBseed*10225210230 G (h50.6520.5) at de- coupling forVL50.7 andVm50.3 @2#. The observations o micro-Gauss magnetic fields in two high-redshift objects~see @1,2# and references therein! could, if correct, impose more stringent conditions on the seeds fields or even on the namo mechanism itself. The cosmological origin of the seed fields is one of t most interesting possibilities, although some other mec nisms at the astrophysical level, such as the Biermann tery process, have also been considered@3,4#. In the cosmo- logical case, in which we will be mainly interested in th work, it is natural to expect@5# that the same mechanism th gave rise to the large-scale galactic structure, i.e. amplifi tion of quantum fluctuations during inflation, was also r sponsible for the generation of the primordial magne fields. However, it was soon noticed@5# that the gravitational amplification does not operate in the case of electromagn ~EM! fields. This is because of the conformal triviality o Maxwell equations in Friedmann-Robertson-Walker~FRW! backgrounds, i.e. conformally invariant equations in a c formally flat space-time. In order to avoid this difficulty, se eral production mechanisms have been proposed in w Maxwell equations are modified in different ways. Thus f example, the addition of mass terms to the photon or hig curvature terms in the Lagrangian was studied in@5#. The contribution of the conformal anomaly was included in@6#. In the context of string cosmology, the effects of a dynami dilaton field were taken into account in@7#. Other examples 0556-2821/2001/64~8!/083006~6!/$20.00 64 0830 r- n d r- e g ry e e e y- a- t- a- - c tic - ch r r- l include non-minimal gravitational-electromagnetic coupli @8#, inflaton coupling to EM Lagrangian@9#, spontaneous breaking of Lorentz invariance@10# or back reaction of mini- mally coupled charged scalars@11–13#. Some of them are able to generate fields of the required strength to seed galactic dynamo or even to account for the observati without further amplification. In this paper we explore the alternative possibility, i.e. w avoid conformal triviality by considering deviations from th FRW metric ~see @14# for a suggestion along these lines!. This approach is rather natural since we know that galax formed from small metric inhomogeneities present at la scales and, in addition, it does not require any modificat of Maxwell electromagnetism. In the inflationary cosmolog metric perturbations are generated when quantum fluc tions become super-Hubble sized and thereafter evolve classical fluctuations, reentering the horizon during radiat or matter dominated eras@15#. The same mechanism woul operate on large-scale EM fluctuations. However, if conf mal invariance is not broken, each positive or negative f quency EM mode will evolve independently, without mixin This implies that photons cannot be created and there magnetic fields are not amplified. However, in the prese of an inhomogeneous background, we will show that mode-mode coupling between EM and metric perturbati generates the mixing. This in turn will allow us to relate t strength of the magnetic field created by this mechanism the particular form of the metric perturbations described the corresponding power spectrum. Those photons produ in the inflation-radiation transition with very long wave lengths can be seen as static electric or magnetic fields. cause of the high conductivity of the Universe in the rad tion era, the electric components are rapidly damp whereas, thanks to magnetic flux conservation, the magn fields will remain frozen in the plasma and their subsequ evolution will be trivial, Ba25const@5,9#. The paper is or- ganized as follows. In Sec. II we obtain the Maxwell equ tions in the presence of an inhomogeneous background calculate the occupation number of the photons produced Sec. III we apply these results to calculate the correspond magnetic field generated at galactic scales. Section IV is voted to the analysis of the effects of finite conductivity those results and finally, Sec. V includes the main conc sions of the paper. ©2001 The American Physical Society06-1 o ck e id e e ns fe ed i- s on w e in that te. und, ar ith ov he a- m- t- ANTONIO L. MAROTO PHYSICAL REVIEW D 64 083006 II. MAXWELL EQUATIONS AND PHOTON PRODUCTION Although there are previous works on the production scalar and fermionic particles in inhomogeneous ba grounds@16,17#, in this paper we will need to extend th analysis to the case of gauge fields. Let us then cons Maxwell equations ¹mFmn50, ~1! in a background metric that can be split asgmn5gmn 0 1hmn , where gmn 0 dxmdxn5a2~h!~dh22d i j dxidxj ! ~2! is the flat FRW metric in conformal time and hmndxmdxn52a2~h!F~dh21d i j dxidxj ! ~3! is the most general form of the linearized scalar metric p turbation in the longitudinal gauge and where it has be assumed that the spatial part of the energy-momentum te is diagonal, as indeed happens in the inflationary or per fluid cosmologies@15#. In this expressionF(h,xW ) is the gauge invariant gravitational potential. Equation~1! can be written as: 1 Ag ] ]xm @Aggmagnb~]aAb2]bAa!#50, ~4! which leads in this background to the following lineariz equations ] ]xi @~122F!~] iA02]0Ai !#50, ~5! for n50 and ] ]h @~122F!~] iA02]0Ai !# 1 ] ]xj @~112F!~] jAi2] iAj !#50, ~6! for n5 i . In addition, we will use the Coulomb gauge cond tion ¹W •AW 50. In order to study the amplification of vacuum fluctuation let us consider a particular solution of the above equati that we will denote byAm kW ,l(x) such that asymptotically in the past it behaves as a positive frequency plane wave momentumkW and polarizationl, i.e. Am kW ,l~x! → h→2` Am (0)kW ,l~x!5 1 A2kV em~kW ,l!ei (kWxW2kh) , ~7! where k25kW2. For the two physical polarization states w have,eW (kW ,l)•kW50 ande0(kW ,l)50. We will work in a finite box with comoving volumeV and we will take the con- tinuum limit at the end of the calculation. We are assum 08300 f - er r- n or ct , s ith g that metric perturbations vanish before inflation starts, so we can define an appropriate initial conformal vacuum sta Because of the presence of the inhomogeneous backgro in the asymptotic future, this solution will behave as a line superposition of positive and negative frequency modes w different momenta and different polarizations, i.e., Am kW ,l~x! → h→` ( l8 ( q S akqll8 em~qW ,l8! A2qV ei (qW xW2qh) 1bkqll8 em* ~qW ,l8! A2qV e2 i (qW xW2qh)D . ~8! It is possible to obtain an expression for the Bogolyub coefficientsakqll8 and bkqll8 to first order in the metric perturbations. With that purpose, we look for solutions of t equations of motion in the form: Am kW ,l~x!5Am (0)kW ,l~x!1Am (1)kW ,l~x!1••• ~9! whereAm (0)kW ,l(x) is the solution in the absence of perturb tions given by Eq.~7!. Introducing this expansion in Eq.~5! and Fourier transforming, we obtain for the temporal co ponent of the EM field to first order in the perturbations: A0 (1)kW ,l~qW ,h!52A2k V qW •eW~kW ,l! q2 F~kW1qW ,h!e2 ikh ~10! where, as usual,F(qW ,h)5(2p)23/2*d3xeiqW xWF(xW ,h). The zeroth order equation impliesA0 (0)kW ,l(qW ,h)50. The spatial equations~6! can be written to first order as: 2F8Ai (0)81] iA0 (1)82Ai (1)912¹W F•¹W Ai (0)22¹W F•] iAW (0) 1¹W 2Ai (1)14F¹W 2Ai (0)50 . ~11! Inserting again expansion~9!, these equations can be rewri ten in Fourier space as: d2 dh2 Ai (1)kW ,l~qW ,h!1q2Ai (1)kW ,l~qW ,h!2Ji kW ,l~qW ,h!50 ~12! where: Ji kW ,l~qW ,h!52A2k V F S iF8~kW1qW ,h! 1 k22kW•qW k F~kW1qW ,h! D e i~kW ,l!e2 ikh 1@eW~kW ,l!•qW #F~kW1qW ,h! ki k e2 ikh 2 i eW~kW ,l!•qW q2 d dh @F~kW1qW ,h!e2 ikh#qi G . ~13! 6-2 h g ho fo la c e a i t in ed r io at - ion e n th er e ar- e of - ing of ted ld ill PRIMORDIAL MAGNETIC FIELDS FROM METRIC . . . PHYSICAL REVIEW D 64 083006 Solving these equations we find, up to first order in t perturbations: Ai kW ,l~qW ,h!5 e i~kW ,l! A2kV d~qW 2kW !e2 ikh 1 1 qEh0 h Ji kW ,l~qW ,h8!sin„q~h2h8!…dh8 , ~14! whereh0 denotes the starting time of inflation. Comparin this expression with Eq.~8!, it is straightforward to obtain the Bogolyubov coefficientsbkqll8 , which are given by: bkqll85 2 i A2qV E h0 h1 eW ~qW ,l8!•JW kW ,l~qW ,h!e2 iqhdh ~15! whereh1 denotes the present time. The total number of p tons created with comoving wave numberkG52p/lG , cor- responding to the relevant coherence length, is there given by @18#: NkG 5 ( l,l8 ( k ubkkGll8u 2 . ~16! We will concentrate only in the effect of super-Hubble sca perturbations whose evolution is relatively simple@15#: F~kW ,h!5Ck 1 a d dh S 1 aE a2dh D1Dk a8 a3 , ~17! where the second term decreases during inflation and soon be neglected. Thus, it will be useful to rewrite the p turbation as:F(kW ,h)5CkF(h). During inflation or preheat- ing, these perturbations evolve in time, whereas they practically constant during radiation or matter eras. We w neglect the effects of the perturbations once they reenter horizon. This is a good approximation for modes reenter right after the end of inflation since they are rapidly damp In addition, we will show that those modes are the mo relevant ones in the calculation. The power spectrum corresponding to Eq.~17! is given by: PF~k!5 k3uCku2 2p2V 5AS 2S k kC D n21 , ~18! where for simplicity we have taken a power-law behav with spectral indexn and we have set the normalization the Cosmic Background Explorer~COBE! scale lC .3000 Mpc withAS.531025. In the case of a blue spec trum, with positive tilt (n.1), perturbations will grow at small scales and it is necessary to introduce a cutoffkmax in order to avoid excessive primordial black hole product @19#. Accordingly, only below the cutoff the perturbativ method will be reliable. For negative tilt or scale-invaria spectrum there will be also a small scale cutoff related to 08300 e - re r an r- re ll he g . e r t e minimum size of the horizonkmax&aIHI , where theI sub- script denotes the end of inflation. We can obtain an explicit expression for the total numb of photons~16! in terms of the power spectrum. Taking th continuum limit(k→(2p)23/2V*d3k, we get: NkG 5 ( l,l8 VE d3k ~2p!3/2 ubkkGll8u 2 5 ( l,l8 VE d3k ~2p!3/2 uCuk1kGuu2 2kGV2 U 3E dhHA2kF S iF81 k22kW•kWG k FD 3@eW~kW ,l!•eW~kWG ,l8!#1@eW~kW ,l!•kWG# 3@eW~kWG ,l8!•kW # F k Ge2 i (kG1k)hJ U2 . ~19! Notice that the last term in Eq.~13! does not contribute to bkqll8 because of the transversality condition of the pol ization vectors. The integration ind3k is dominated by the upper limit, i.e.k@kG and accordingly we can ignore th effect of the terms proportional tokWG . In addition, for those modesk which are outside the Hubble radius at the end inflation, we havekh!1. With these simplifications we ob tain: NkG . ( l,l8 E dk dV ~2p!3/2 uCku2k2 2kGV 3U E dh„A2k$~ iF81kF!@eW~kW ,l!•eW~kWG ,l8!#%…U2 . ~20! Performing the integration in the angular variables and us the definition of the power spectrum in Eq.~18!, we obtain: NkG . 4~2p!3/2 3kG E dkAS 2S k kC D n21U E dh~ iF81kF!U2 . ~21! Finally, we will estimate the time integral. The behavior scales that reenter the horizon during the radiation domina era is oscillatory with a decaying amplitude@15#, therefore, there is no long-time contribution to the integral that cou spoil the perturbative method. Thus, for simplicity we w assume that the functionF vanishes forh>1/k, and accord- ingly we estimate,u*dh( iF81kF)u2;O(1). Our final ex- pression for the occupation number is: NkG . 4~2p!3/2AS 2 3kG~kC!n21EkC kmax dkkn21. 4~2p!3/2AS 2 3 n kmax n kG kC n21 . ~22! 6-3 c- at ti re th er ce b rr m n r io n ge un ca a t i s, tia re ti - d w the re- ctric ion a e- co- the g d g n ld c- ay set een o- ire ro- del, in he vo- ich actic os- ANTONIO L. MAROTO PHYSICAL REVIEW D 64 083006 III. MAGNETIC FIELD GENERATION The energy density stored in a magnetic field modeBk with wave numberk is given by: rB~v!5v drB dv 5 uBku2 2 , ~23! with v5k/a the physical wave number. In terms of the o cupation number it reads rB~v!5v4Nk . ~24! From Eq. ~22!, we can obtain the strength of the field decoupling on a coherence scale corresponding tokG ;10236 GeV as: uBkG decu.A2~vG dec!2NkG 1/2. 23/2~2p!3/4AS A3n adec 2 kmax n/2 kG 3/2 kC (n21)/2 . ~25! In Fig. 1 we have plotted the strength of the magne field generated as a function of the comoving cutoff f quencykmax for different values of the spectral indexn. No- tice that the results are in general too weak to explain observed fields without any amplification. However, for c tain values of the cosmological parameters, the produ fields could act as seeds for a galactic dynamo. We see that the spectrum of magnetic fields produced this mechanism is thermalBk;k3/2, in the low-momentum region. We can then compare this spectrum with that co sponding to the thermal background radiation with a te peratureTdec.0.26 eV present at decoupling time. The e ergy density in photons with comoving wave numberkG at decoupling is given byrR(vG).kG 3 Tdec/adec 3 . Thus we find: rR~vG! rB~vG! 5 adecTdec NkG kG .1.431036S kC kmax D n . ~26! From this expression we see that the magnetic field ene density will dominate over the background thermal radiat whenever log(kmax/kC)*36/n, i.e. for example, forn51 this implies kmax*1026 GeV. The cutoff frequencykmax cannot be easily determined i general, since it depends on the specific mechanism that erates the perturbations and also on the evolution of the verse during reheating and thermalization. However we estimate typical values in some particular regime. In the c in which metric perturbations are generated by inflation, i natural to expect,kmax&aIHI , as commented before. Thu let us take the simplest chaotic inflation model with poten V(f)5lf4/4 @20#, with l.10212 fixed by COBE. In this model the Hubble parameter during inflation isHI .1013 GeV. Owing to the uncertainties commented befo we will let the reheating temperatureTRH be a free param- eter. After reheating the universe evolution is adiaba aI /adec;Tdec/TRH , and we can calculate the cutoff fre quency askmax/kC;aIHI /kC;adecTdecHI /(TRHkC), which yields kmax/kC;1042 GeV/TRH . Here we have assume 08300 c - e - d y e- - - gy n n- i- n se s l , c that the inflation-radiation transition takes place in a fe inflaton oscillations@20# ~see also@21#!. Comparing with Fig. 1 we see that with these simple estimations for lf4/4 model, the amplification could be above the requi ments of the galactic dynamo ifTRH&106 GeV for n 51.25. IV. CONDUCTIVITY EFFECTS In the previous discussion we have assumed that ele conductivity of the universe played no role in the generat of the magnetic fields. Although neglecting conductivity is good approximation during inflation, it is not during the r heating or radiation periods. As commented before, the pious production of particles during reheating produces growing of the conductivity which becomes very high durin radiation@5#. This implies that the magnetic fields produce in the inflation-radiation transition will evolve conservin magnetic fluxrB;a24. However, it has been recently show @13# that the growth of conductivity during reheating cou affect the evolution of the EM modes. The effects of condu tivity can be taken into account in a phenomenological w by introducing a current sourceJi5scaAi8 in Eq. ~11!. This approach is only valid at sufficiently large scales@13# and in general the rigorous treatment would require to solve the of coupled EM-matter fields equations~Vlasov equations! which is beyond the scope of this paper. In general,sc(xW ,h) is a time and space dependent function, and it has b shown that it grows exponentially during parametric res nance in a non-equilibrium plasma in QED@22#. The calcu- lation of the actual functionsc(xW ,h) in our case would be rather involved and model dependent, since it would requ information about the reheating and thermalization p cesses. For that reason we will not take any particular mo but we will do a general discussion of the possible effects different cases. We will also assume for simplicity that all t Fourier modes of the conductivity have the same time e lution, i.e.sc(kW ,h)5S(h)sk during reheating. FIG. 1. log(BkG dec/1 G) as a function of log(kmax/kC). The con- tinuous line corresponds to the scale-invariant Harrison-Zeldov spectrum withn51, the dashed line ton50.8 and the dotted line to n51.25. The dashed horizontal line represents the weakest gal dynamo seed field limit corresponding to a flat universe with c mological constant andh50.5. 6-4 - ac ne o io de ill y: on - - tio t a e me mi- i- this cts des in und. ce he nd tric ism ctic pli- ved ical cha- nt red we M form he ns, e lds er- les rio PRIMORDIAL MAGNETIC FIELDS FROM METRIC . . . PHYSICAL REVIEW D 64 083006 In previous works@12,13#, the conductivity was consid ered as a homogeneous field,sc(h) and because of the form of the current source term, its effect was the damping of e EM mode. In our production mechanism, the inhomoge ities are able to mix different EM modes and for this reas we need to have information about the complete spectrumsk and not only about the large-scale components. In addit our analysis is perturbative and therefore we can only scribe the initial stages in which the conductivity is st small. Let us then consider the modified spatial equation first order in the metric perturbations and the conductivit ~2F82asc!Ai (0)81] iA0 (1)82Ai (1)912¹W F•¹W Ai (0) 22¹W F•] iAW (0)1¹W 2Ai (1)14F¹W 2Ai (0)50 . ~27! Following a similar analysis we find the same Eq.~12!, but with the new current: Ji kW ,l~qW ,h!52A2k V F S iF8~kW1qW ,h!2 i 2 a sc~kW1qW ,h! 1 k22kW•qW k F~kW1qW ,h! D e i~kW ,l!e2 ikh 1@eW~kW ,l!•qW #F~kW1qW ,h! ki k e2 ikh 2 i eW~kW ,l!•qW q2 d dh @F~kW1qW ,h!e2 ikh#qi G . ~28! In @13# the following lower limit on the~homogeneous! con- ductivity is obtainedaI sc;aI HI /a, with a the fine struc- ture constant, i.e.aI sc;kmax/a and this implies that the conductivity term will dominate in Ji kW ,l(kWG ,h) for k !kmax. However, as commented before, the dominant c tribution to the EM amplification comes from the high frequency modes, i.e.k;kmax. In such a case, the impor tance of the conductivity term is determined by the ra sk /Ck when k→kmax. Thus, if the conductivity is almos homogeneous, its spectrum will decline at short scales . J ev 08300 h - n n, - to - nd we expect its contribution to the EM field evolution to b negligible. In the opposite case, the analysis would beco much more involved, since the above magnetohydrodyna cal approximation would break down and the full set of m croscopic equations would be needed. In any case, simple analysis shows that the possible damping effe mainly affect those modes withk;kG and that, depending on the actual conductivity spectrum, the rest of the mo could be less severely affected than in other models. V. CONCLUSIONS In this work we have studied the production of photons the presence of an inhomogeneous gravitational backgro We have shown how the breaking of conformal invarian induced by the evolution of metric perturbations in t inflation-radiation transition is able to produce particles, a we have related the occupation number with the scalar me perturbations power spectrum. We have considered the possibility that this mechan could have had some relevance in the problem of gala magnetic fields and we have concluded that the total am fication is several orders of magnitude below the obser strengths. However, for certain values of the cosmolog parameters and with the assistance of the dynamo me nism, the amplification could be compatible with the curre ~low-redshift! galactic observations. We have also conside the effect of conductivity in a phenomenological way and showed that although it could affect the evolution of E modes, in some cases and depending on the particular of the spectrum, the effects could be small. The mechanism studied in this work only relies on t existence of a primordial spectrum of metric perturbatio described by the scalar spectral indexn and a possible cutoff frequencykmax, which are the only free parameters in th model. Therefore, as a by-product we get that magnetic fie could also provide useful information about the metric p turbations spectrum and in particular about its small-sca region. ACKNOWLEDGMENTS This work has been partially supported by the Ministe de Educacio´n y Ciencia~Spain! ~CICYT AEN 97-1693!. ys. ys. @1# P.P. Kronberg, Rep. Prog. 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